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  1. 019A plant's output grows 10% every month. Roughly how many months until output doubles, and how long until it is eight times today's level? Use the rule of 72, then check it against the exact answer.Compounding and time valueWarm upConsulting-style caseCorporate FP&A

    Try it first

    How long until output is eight times today's level?

    Show the worked solution

    Output doubles in about 7.2 months and reaches eight times in about 21.6 months, by the rule of 72. Divide 72 by the growth rate in per cent: 72 over 10 is 7.2 months per doubling, and eight times is three doublings. The exact answer, the log of 2 over the log of 1.1, is 7.27 months, or 21.8 months for eight times, so the rule is within a fifth of a month.

    Why does eight times take three doublings and not 70 months?

    Think of a rumour that doubles the number of people who know it every week. One becomes two, then four, then eight in three weeks, not eight weeks. Compounding growth multiplies, so the question to ask is how many doublings fit in, not how many 10% steps add up to 700%. Two times two times two is eight, so eight times today's output is three doublings away. Dividing 700% by 10% a month treats each month's growth as if it were earned only on the starting level.

    The relationship
    t2x=ln⁡2ln⁡1.10=0.6930.0953=7.27rule of 72: 7210=7.2t_{2x} = \frac{\ln 2}{\ln 1.10} = \frac{0.693}{0.0953} = 7.27 \qquad \text{rule of 72: } \frac{72}{10} = 7.2
    ln 2the natural log of 2, about 0.693
    ln 1.10the natural log of one month's growth factor, about 0.0953
    72a convenient number close to 100 times ln 2, with many divisors
    What it says in wordsDoubling time is the log of two divided by the log of one period's growth factor; the rule of 72 approximates that with one division.
    Output at 10% a month: three doublings in about 22 months2x4x8x1x06121824MonthsOutput, multiple of todayexact 7.27rule of 72: 7.2exact 14.55rule of 72: 14.4exact 21.82rule of 72: 21.6
    At 10% a month, output reaches twice today's level after 7.27 months, four times after 14.5 and eight times after 21.8. The rule of 72 marks at 7.2, 14.4 and 21.6 months sit just short of each, so the shortcut is close enough to say out loud.

    Where does the rule of 72 work, and where does it slip?

    The exact doubling time uses the log of 1 plus the rate, which is a little less than the rate itself, and 72 is chosen to compensate for typical rates. The rule is within about 1% for rates from 5% to 10% a period and drifts at the extremes: at 2% it overstates, and at 50% it understates. At 10% the gap is only 0.07 months. A check in whole months helps too: 1.1 to the 7th is 1.95 and to the 8th is 2.14, so output passes double during month 8.

    Growth per periodRule of 72ExactGap
    2%36.0035.00+2.8%
    5%14.4014.21+1.4%
    10%7.207.27-1.0%
    25%2.883.11-7.3%
    50%1.441.71-15.8%
    Periods to double, with the rule's error as a share of the exact answer. It is within about 1% at 5% and 10% a period, and drifts further at very low and very high rates.

    Say the limit in context. Output growing 10% a month for two years is a startup or a ramp-up, not a steady state; a plant hits capacity, demand or cash constraints long before it has grown nearly tenfold. The interviewer wants the arithmetic, then one sentence of business sense.

    Where candidates lose it

    The common loss is answering about 70 months for eight times, treating the growth as simple rather than compound. It is the same instinct that underestimates how quickly compounding builds.

    The second loss is reaching for a calculator-style log in your head and stalling. Give the rule of 72 answer first, then say the exact number is a little longer, about 7.3 months.

    What the interviewer asks next

    • Using the same idea, how long does it take for prices to halve in value at 6% inflation?
    • Why do some people use 69 or 70 instead of 72?
    • Output grows 10% a month for a year. What is the annual growth rate?
  2. 070One lender quotes 12% a year compounded monthly; another quotes 12.5% a year compounded annually. Which loan is cheaper, and what is the effective annual rate of each?Compounding and time valueWarm upCarlyle GroupNew York · 2015

    Try it first

    Which loan is cheaper?

    Show the worked solution

    The 12.5% annual loan is cheaper. Twelve per cent compounded monthly is an effective 12.68% a year, against 12.50%. A 12% nominal rate charged monthly means 1% a month, and interest earns interest eleven more times within the year: 1.01 to the power 12, less 1, is 12.68%. Convert every quote to an effective annual rate before comparing.

    Why is 12% a year compounded monthly more than 12%?

    Leave a credit card balance unpaid and the interest charged in January is itself charged interest in February. The bank's 'monthly rate' grows faster than the same rate charged once a year. A nominal rate tells you how interest is quoted; the effective annual rate tells you what it costs, and only effective rates can be compared. Here 12% a year becomes 1% a month, and after twelve months of compounding Rs 100 has grown to Rs 112.68.

    Same 12% headline, five different costs: convert before you compare12.0%12.4%12.8%Lender B: 12.50% compounded annually12.00%Annual12.36%Half-yearly12.55%Quarterly12.68%Monthly12.75%DailyLender A: 12% monthlyCompounding frequency of a 12% nominal rate
    The same 12% nominal rate costs anything from 12.00% to about 12.75% a year depending on how often it compounds, and at monthly compounding its 12.68% sits above the other lender's 12.50%, so the annual quote is cheaper.
    The relationship
    EAR=(1+0.1212)12−1=1.0112−1≈12.68%  >  12.50%\text{EAR} = \left(1 + \frac{0.12}{12}\right)^{12} - 1 = 1.01^{12} - 1 \approx 12.68\% \;>\; 12.50\%
    0.12the nominal annual rate
    12compounding periods a year
    EAReffective annual rate, the true yearly cost
    What it says in wordsDivide the nominal rate by the number of periods, compound it over a year, and subtract one to get the rate you can compare.
    Compounding of 12% nominalEffective annual rate
    Annual12.00%
    Half-yearly12.36%
    Quarterly12.55%
    Monthly12.68%
    Daily12.75%
    Continuous12.75%
    More frequent compounding raises the effective rate, but with sharply diminishing steps: going from monthly to continuous adds less than a tenth of a point.

    What would the monthly lender need to quote to match?

    Run the conversion backwards: the monthly rate that compounds to 12.5% is 1.125 to the power 1/12, less 1, about 0.98% a month, or a nominal 11.84% a year. Any monthly quote above 11.84% is dearer than 12.5% annual. For a quick mental check, the extra from monthly compounding at these rates is roughly half the rate squared: 0.5 x 0.12 x 0.12 is 0.72 points, close to the true 0.68.

    What else decides which loan is really cheaper?

    The effective rate prices the money, not the whole deal. Processing fees, prepayment penalties and insurance bundled into the loan all add to the cost and must be folded in. Watch for flat-rate quotes too: a 7% flat rate on a three-year loan charges interest on the original amount even as it is repaid, which works out to an effective rate of about 13.6% a year. Confirm how any lender computes its rate before comparing; the limit of the EAR is that it compares like with like only once every cost is in it.

    Where candidates lose it

    The fast wrong answer picks 12% because it is the smaller number. It compares quotes built on different compounding, which is comparing prices in two currencies without converting.

    The other slip is the reverse: knowing that compounding adds cost but guessing the size. Monthly compounding adds about 0.68 points at 12%, not one or two points, so a 12.5% annual quote only just wins. Compute it rather than estimate it.

    What the interviewer asks next

    • What is the effective annual rate of 1.5% a month on a credit card?
    • A deposit pays 7% compounded quarterly. What is its effective annual yield?
    • Why does continuous compounding at 12% give about 12.75%, and what function produces it?

    Asked at Carlyle Group, Generalist, New York, 2015 (Wall Street Oasis): Some math brainteasers and accounting questions ranging from compounding rates to how an inventory purchase would flow